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Construction of periodic solutions to partial differential equations with nonlinear boundary conditions. (English) Zbl 0977.35031

From the introduction: We consider the following equation governing the behaviour of waves \[ u_{tt}=u_{xx}\tag{1} \] with the following boundary conditions \[ u(0,t)=0,\tag{2} \]
\[ u_x(1,t)+u(1,t)+\alpha u^3(1,t)=0.\tag{3} \] A similar problem, for \(|\alpha|\ll 1\), can be efficiently solved by means of the perturbation technique. However, if \(a\sim 1\), then a problem becomes considerably difficult. If a solution to the boundary problem (1)–(3) is sought in the form \[ u=\sum^\infty_{j=1,3,5,\dots} A_j\sin\frac{\pi j x}{2}\sin\frac{\pi jt}{2} , \] then one obtains an infinite set of nonlinear algebraic equations with the unknowns \(A_j\).
In the works of C. M. Bender, K. A. Milton, S. S. Pinsky and L. M. Simmons jun. [J. Math. Phys. 30, No. 7, 1447-1455 (1989; Zbl 0684.34008)] and C. M. Bender, S. Boettcher and K. A. Milton [J. Math. Phys. 32, No. 11, 3031-3038 (1991; Zbl 0741.35064)] the so called small \(\delta\) method has been proposed. A small artificial \(\delta\) parameter is introduced in the power exponent of a nonlinear term. According to that approach the boundary condition (3) can be formulated in the following manner \[ u_x+u+\alpha u^{1+2\delta}=0.\tag{4} \] A solution to the boundary problem governed by (1), (2), (4) is sought as a series of the parameter \(\delta\).
An application of the above mentioned procedure to a series of nonlinear equations shows its high efficiency. In this work we are going to apply it for construction of periodic solutions of the boundary-value problem governed by (1), (2) and (4).

MSC:

35C10 Series solutions to PDEs
35B10 Periodic solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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References:

[1] Szemplinska-Stupnicka W., International Journal of Non-Linear Mechanics 21 (5) pp 401– (1986) · Zbl 0628.70020 · doi:10.1016/0020-7462(86)90023-5
[2] Awrejcewicz J, Meccanica 31 pp 347– (1996) · Zbl 0870.70015 · doi:10.1007/BF00426995
[3] He J. H, International Journal of Nonlinear Sciences and Numerical Simulation 1 (1) pp 51– (2000)
[4] Andrianov I. V., Technische Mechanik 15 (1) pp 53– (1995)
[5] Bender C. M., Journal of Mathematical Physics 30 (7) pp 1447– (1989) · Zbl 0684.34008 · doi:10.1063/1.528326
[6] Bender C. M, Journal of Mathematical Physics 32 (11) pp 3031– (1991) · Zbl 0741.35064 · doi:10.1063/1.529047
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